Internat. J. Math. & Math. Sci.
Vol. 23, No. 9 (2000) 639–644
S0161171200002234
© Hindawi Publishing Corp.
ELEMENTS IN EXCHANGE RINGS WITH RELATED
COMPARABILITY
HUANYIN CHEN
(Received 23 December 1998)
Abstract. We show that if R is an exchange ring, then the following are equivalent: (1)
R satisfies related comparability. (2) Given a, b, d ∈ R with aR + bR = dR, there exists a
related unit w ∈ R such that a + bt = dw. (3) Given a, b ∈ R with aR = bR, there exists a
related unit w ∈ R such that a = bw. Moreover, we investigate the dual problems for rings
which are quasi-injective as right modules.
Keywords and phrases. Exchange ring, related comparability, related unit.
2000 Mathematics Subject Classification. Primary 16E50, 16L99.
Let R be an associative ring with identity. From [6], R is said to satisfy related comparability provided that for any idempotents e, f ∈ R with e = 1 + ab and f = 1 + ba
for some a, b ∈ R, there exists a u ∈ B(R) such that ueR ⊕ uf R and (1 − u)f R ⊕
(1−u)eR. The class of rings satisfying related comparability is quite large. It includes
regular rings satisfying general comparability [10], one-sided unit regular rings [8] and
partially unit-regular rings, while there still exist rings satisfying related comparability, which belong to none of the above classes (cf., [7, Example 10]).
In [4, 5], we studied related comparability over regular rings. In [6, 7], we investigated related comparability over exchange rings. It is shown that every exchange ring
satisfying related comparability is separative [1]. Also, we show that related comparability over exchange rings is a Morita invariant. R is said to be an exchange ring if
for every right R-module A and any two decompositions A = M ⊕ N = i∈I Ai , where
MR R and the index set I is finite, then there exist submodules Ai ⊆ Ai such that
A = M ⊕ ( i∈I Ai ). Many authors have investigated exchange rings with some kind of
comparability properties so as to study problems related partial cancellation properties of modules (see [1, 2, 6, 7, 12, 13]).
In this paper, we investigate related comparability over exchange rings by related
units. Recall that w ∈ R is said to be a related unit of R if there exists some e ∈ B(R)
such that w = eu + (1 − e)v for some u, v ∈ R, where eu is right invertible in eR and
(1 − e)v is left invertible in (1 − e)R. w ∈ R is said to be a semi-related unit of R if
w ∈ R is a related unit modulo J(R). By virtue of semi-related units, we also give some
new element-wise properties of rings which are quasi-injective as right modules.
Throughout, all rings are associative with identities. B(R) denotes the set of all
central idempotents of R and r · ann(b)(l · ann(b)) denotes the right (left) annihilator
of b ∈ R.
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HUANYIN CHEN
Lemma 1. Let R be an exchange ring. Then R satisfies related comparability if and
only if so does the opposite ring R op of R.
Proof. Since R is an exchange ring, by virtue of [11, Proposition], so is the opposite
ring R op of R. Assume that R satisfies related comparability. Given aop , bop ∈ R op with
aop x op +bop = 1op , then we have xa+b = 1 in R. In view of [6, Theorem 4], there exists
a y ∈ R such that x +by is a related unit of R. Thus, we have some e ∈ B(R) such that
(x + by)e is right invertible in eR and (x + by)(1 − e) is left invertible in (1 − e)R. By
[5, Lemma 4], we claim that there are z1 , z2 ∈ R such that (a + z1 b)e is left invertible
in eR and (a + z2 b)(1 − e) is right invertible in (1 − e)R. Let z = z1 e + z2 (1 − e). Then
a + zb is a related unit of R. Consequently, aop + bop zop is a related unit of R op . By
[6, Theorem 4], we conclude that R op satisfies related comparability. The converse is
clear from R (R op )op .
Theorem 2. Let R be an exchange ring. Then the following are equivalent:
(1) R satisfies related comparability.
(2) Given a, b, d ∈ R with aR + bR = dR, there exists a related unit w ∈ R such that
a + bt = dw.
(3) Given a, b with aR = bR, there exists a related unit w ∈ R such that a = bw.
(4) Given a, b, d ∈ R with Ra + Rb = Rd, there exists a related unit w ∈ R such that
a + tb = wd.
(5) Given a, b with Ra = Rb, there exists a related unit w ∈ R such that a = wb.
Proof. (2) ⇒(1). Trivial from [6, Theorem 4].
(1) ⇒(2). Given a, b, d ∈ R with aR + bR = dR. Let g : dR → dR/bR be the canonical
map, f1 : R → aR given by r ar for any r ∈ R, f2 : R → bR given by r br for
any r ∈ R, f3 : R → dR given by r dr for any r ∈ R. Since aR + bR = dR, we know
that gf1 , gf3 are epimorphisms. On the other hand, R is a projective R-module. So
there is some α ∈ EndR R such that gf1 = gf3 α. Since gf1 is a epimorphism, we also
have some ψ ∈ EndR R such that gf3 αψ = gf3 . From αψ + (1 − αψ) = 1, there is
a y ∈ EndR R such that α + (1 − αψ)y = w is a related unit of EndR R. Therefore,
we see that gf1 = gf3 α = gf3 (α + (1 − αψ)y) = gf3 w, and then g(f1 − f3 w) = 0.
Thus, we have Im(f1 − f3 w) ≤ Ker g = bR. By the projectivity of right R-module R,
there exists some β ∈ EndR R such that f2 β = f1 − f3 w. Therefore, we claim that
a + bβ(1) = f1 (1) + f2 (1)β(1) = f3 (1)w(1) = dw(1). It is easy to verify that w(1) is a
related unit of R.
(1) ⇒(3). Given a, b ∈ R with aR = bR, there exist s, t ∈ R such that a = bs and
b = at. Thus, b = bst. Since st + (1 − st) = 1, by virtue of [6, Theorem 4], there exists
some z ∈ R such that s + (1 − st)z = w is a related unit of R. Hence a = bs = b(s +
(1 − st)z) = bw, as desired.
(3) ⇒(1). Given any regular a ∈ R. Then there exists some b ∈ R such that a = aba,
so aR = abR. Thus a = abw for some related unit w ∈ R. Since ab + (1 − ab) = 1, we
see that a + (1 − ab)w = (ab + (1 − ab))w = w. By [5, Lemma 4], there is some z ∈ R
such that b+z(1−ab) = m is a related unit of R. Hence a = aba = a(b+z(1−ab))a =
ama. According to [6, Theorem 2], we claim that R satisfies related comparability.
(1)⇐⇒(4)⇐⇒(5). By [11, Proposition], we see that the opposite ring R op of R is
ELEMENTS IN EXCHANGE RINGS WITH RELATED COMPARABILITY
641
exchange. Using Lemma 1, we see that R satisfies related comparability if and only
if so does the opposite ring R op of R. Applying (1)⇐⇒(2)⇐⇒(3). To R op , we easily derive the result.
Corollary 3. Let R be an exchange ring. Then the following are equivalent:
(1) R satisfies related comparability.
(2) Given a, b ∈ R with aR + r · ann(b) = R, there exists some k ∈ r · ann(b) such that
a + k is a related unit.
(3) Given a, b ∈ R with Ra + l · ann(b) = R, there exists some k ∈ l · ann(b) such that
a + k is a related unit.
Proof. (1) ⇒(2). Given a, b ∈ R with aR +r · ann(b) = R, then there exist x ∈ R, k ∈
r · ann(b) such that ax + k = 1. Since R satisfies related comparability, by virtue of [6,
Theorem 4], we can find a y ∈ R such that a + ky is a related unit of R. It is easy to
check that ky ∈ r · ann(b), as required.
(2) ⇒(1). Given a, b ∈ R with aR = bR, there exist s, t ∈ R such that a = bs and
b = at. Obviously, 1−st ∈ r · ann(b). Since st +(1−st) = 1, we have sR +r · ann(b) = R.
Thus we can find some k ∈ r · ann(b) such that s + k = w is a related unit of R, and
then a = bs = b(s + k) = bw, as asserted.
(1)⇐⇒(3). Trivial by the symmetry of related comparability.
Recall that n is in the stable range of R provided that a1 R + · · · + an+1 R = R with
a1 , . . . , an+1 ∈ R implies that (a1 + an+1 b1 )R + · · · + (an + an+1 bn )R = R for some
b1 , . . . , bn ∈ R. If no such n exists, we say the stable range of R is ∞. x ∈ R is said to
be related unit-regular if x = xwx for some related unit w ∈ R. Now, we investigate
related comparability by related unit-regularity as follows.
Proposition 4. Let R be an exchange ring with the finite stable range. Then the
following are equivalent:
(1) R satisfies related comparability.
(2) Given a, b, d ∈ R with aR+bR = dR, there exist some related unit-regular w1 , w2 ∈
R such that aw1 + bw2 = d.
(3) Given a, b, d ∈ R with Ra+Rb = Rd, there exist some related unit-regular w1 , w2 ∈
R such that w1 a + w2 b = d.
Proof. (1) ⇒(2). Given aR + bR = dR with a, b, d ∈ R. For right R-module R 2 ,
the two sets {a, b} and {0, d} generate the same right R-submodule of R 2 . Thus,
we can find A, B ∈ M2 (R) such that (a, b) = (0, d)A, (0, d) = (a, b)B. Assume that
A = (aij ), B = (bij ), I2 − AB = (cij ) ∈ M2 (R). Since AB + (I2 − AB) = I2 , we have
(a21 , a22 )(b12 , b22 )T + c22 = 1. Since R is an exchange ring satisfying related comparability, its stable range can only be 1, 2 or ∞ by [7, Theorem 3]. So 2 is in the stable
range of R. Thus, we have some (y1 , y2 ) ∈ R 2 such that (a21 , a22 )+c22 (y1 , y2 ) ∈ R 2 is
unimodular. Set Y = y01 y02 . Then, we claim that the second row of A+(I2 −AB)Y = U
is unimodular. Clearly, (0, d)U = (0, d)A = (a, b). Since u21 R + u22 R = R, we can find
orthogonal idempotents e1 ∈ u21 R, e2 ∈ u22 R such that e1 + e2 = 1. Assume that
e1 = u21 x1 , e2 = u22 x2 . Let w1 = x1 e1 , w2 = x2 e2 . Then w1 and w2 are both regular in
R. Moreover, we have u21 w1 + u22 w2 = 1. By the related comparability of R, we claim
that both wi are related unit-regular, as asserted.
642
HUANYIN CHEN
(2) ⇒(1). Given any regular x ∈ R. Then x = xyx for a y ∈ R. So we have xR + (1 −
xy)R = R, and then xw1 + (1 − xy)w2 = 1 for some related unit-regular w1 , w2 ∈ R.
We easily check that x + (1 − xy)w2 s ∈ R is related unit for some s ∈ R. Hence y +
t(1 − xy) = w, i.e., a related unit of R. Consequently, we show that x = xyx = xwx,
as desired.
(1)⇐⇒(3). Clear from the symmetry of related comparability.
Recall that a module M is quasi-injective if any homomorphism of a submodule of
M into M extends to an endomorphism of M. Now, we investigate rings which are
quasi-injective as right modules. These extend the corresponding results in [3].
Lemma 5. Let R be quasi-injective as a right R-module. Given a, b ∈ R with aR +
bR = R, there exists some t ∈ R such that a + bt is a semi-related unit.
Proof. Given a, b ∈ R with aR + bR = R, then ā(R/J(R)) + b̄(R/J(R)) = R/J(R).
Since R is quasi-injective as a right R-module, by virtue of [9, Theorem 1], R/J(R)
is a regular, right self-injective ring. Hence R is an exchange ring satisfying related
comparability. According to Theorem 2, we can find a y ∈ R such that ā + b̄ȳ = w̄ is
a related unit of R/J(R). Therefore a + by = w + r for some r ∈ J(R). Clearly, w + r
is a semi-related unit of R, as desired.
Theorem 6. Let R be quasi-injective as a right R-module. Then the following hold:
(1) Given a, b ∈ R with r · ann(a) = r · ann(b), there exists a semi-related unit w ∈ R
such that a = wb.
(2) Given a, b ∈ R with l · ann(a) = l · ann(b), there exists a semi-related unit w ∈ R
such that a = bw.
Proof. (1) Given a, b ∈ R with r · ann(a) = r · ann(b). Since R is quasi-injective as
a right R-module, by [3, Lemma 3.2], we have Ra = Rb. Assume that a = sb, b = ta
for some s, t ∈ R. Then b = tsb. Consequently, there exists some y ∈ R such that
t + (1 − ts)y is a semi-related unit of R by Lemma 5. Using [5, Lemma 4], we have
some z ∈ R such that s +z(1−ts) = w is a semi-related unit of R. Therefore, we claim
that a = sb = (s + z(1 − ts))b = wb, as desired.
(2) Given a, b ∈ R with l · ann(a) = l · ann(b). Similarly to the consideration above,
we have aR = bR. Assume that a = bs, b = at for some s, t ∈ R. Then b = bst. From
st + (1 − st) = 1, we can find a y ∈ R such that s + (1 − st)y = w is a semi-related unit
of R. Therefore a = bs = b(s + (1 − st)y) = bw, whence the result.
Corollary 7. Let R be quasi-injective as a left R-module. Then the following hold:
(1) Given a, b ∈ R with r · ann(a) = r · ann(b), there exists a semi-related unit w ∈ R
such that a = wb.
(2) Given a, b ∈ R with l · ann(a) = l · ann(b), there exists a semi-related unit w ∈ R
such that a = bw.
Proof. Applying Theorem 6 to the opposite ring R op of R, we complete the proof.
Theorem 8. Let R be a ring which is quasi-injective as a right R-module. Then the
following hold:
ELEMENTS IN EXCHANGE RINGS WITH RELATED COMPARABILITY
643
(1) Given a, b ∈ R with r · ann(a) ∩ r · ann(b) = 0, there exists t ∈ R such that a + tb
is a semi-related unit.
(2) Given a, b ∈ R with l · ann(a) ∩ l · ann(b) = 0, there exists t ∈ R such that a + bt is
a semi-related unit.
Proof. (1) Given a, b ∈ R with r · ann(a) ∩ r · ann(b) = 0, by virtue of [3, Proposition 3.4], we know that Ra+Rb = R. Thus, (R/J(R))ā+(R/J(R))b̄ = R/J(R). Since R is
a quasi-injective ring, from [9, Theorem 1], R/J(R) is a regular, right self-injective ring.
Moreover, we see that R/J(R) satisfies related comparability. In view of Theorem 2,
there exists t ∈ R such that ā + t̄ b̄ = w̄ with w is a semi-related unit of R. Thus, there
is some k ∈ J(R) such that a + tb = w + k. Clearly, w + k is also a semi-related unit.
Thus, we claim that a + tb is a semi-related unit of R.
(2) Given a, b ∈ R with l · ann(a) ∩ l · ann(b) = 0, analogously to [3, Proposition 3.4],
we claim that aR + bR = R. Thus ā(R/J(R)) + b̄(R/J(R)) = R/J(R). Similarly to the
consideration above, we show that R/J(R) satisfies related comparability. In view of
Theorem 2, there exists t ∈ R such that a + bt = w + k with w is a semi-related unit
and k ∈ J(R). Since w + k is also a semi-related unit, the result follows.
Corollary 9. Let R be a ring which is quasi-injective as a left R-module. Then the
following hold:
(1) Given a, b ∈ R with r · ann(a) ∩ r · ann(b) = 0, there exists t ∈ R such that a + tb
is a semi-related unit.
(2) Given a, b ∈ R with l · ann(a) ∩ l · ann(b) = 0, there exists t ∈ R such that a + bt is
a semi-related unit.
Proof. Applying Theorem 8 to the opposite ring R op of R, we easily obtain the
result.
Since every regular, right (left) self-injective ring is a quasi-injective ring with trivial
Jacobson. As an immediate consequence of Theorem 6, Corollary 7, Theorem 8, and
Corollary 9, we now derive the following.
Corollary 10. Let R be a regular, right (left) self-injective ring. Then the following
hold:
(1) Given a, b ∈ R with r · ann(a) = r · ann(b), there exists a related unit w ∈ R such
that a = wb.
(2) Given a, b ∈ R with l · ann(a) = l · ann(b), there exists a related unit w ∈ R such
that a = bw.
(3) Given a, b ∈ R with r · ann(a) ∩ r · ann(b) = 0, there exists t ∈ R such that a + tb
is a related unit.
(4) Given a, b ∈ R with l · ann(a) ∩ l · ann(b) = 0, there exists t ∈ R such that a + bt is
a related unit.
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Chen: Department of Mathematics, Hunan Normal University, Changsha 410006,
China
E-mail address: chyzxl@sparc2.hunnu.edu.cn